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Generalized nonbinary sequences with perfect autocorrelation, flexible alphabets and new periods

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Abstract

We extend the parameters and generalize existing constructions of perfect autocorrelation sequences over complex alphabets. In particular, we address the PSK+ constellation (Boztaş and Udaya 10) and present an extended number theoretic criterion which is sufficient for the existence of the new sequences with perfect autocorrelation. These sequences are shown to exist for nonprime alphabets and more general lengths in comparison to existing designs. The new perfect autocorrelation sequences provide novel alternatives for wireless communications and radar system designers for applications in ranging and synchronisation as well as channel identification.

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References

  1. Boztaş, S., Udaya, P.: Nonbinary Sequences with Perfect and Nearly Perfect Autocorrelations. In: Proceedings of the IEEE International Symposium on Information Theory, pp. 1300–1305. Austin, Texas (2010)

  2. Boztas, S., Kahraman, S., Ozbudak, F., Tekin, E.: A generalized construction for perfect autocorrelation sequences. In: Proceedings of the IEEE International Symposium on Information Theory, pp. 14–19. Hong Kong, China (2015)

  3. Chu, D.C.: Polyphase codes with good periodic correlation properties. IEEE Trans. Inf. Theory IT-18, 531–532 (1972)

    Article  MATH  Google Scholar 

  4. Fan, P.Z., Darnell, M., Honary, B.: Crosscorrelations of Frank sequences and Chu sequences. Electron. Lett. 30(6), 477–478 (1994)

    Article  Google Scholar 

  5. Frank, R.L., Zadoff, S.A.: Phase shift pulse codes with good periodic correlation properties. IRE Trans. Inf. Theory IT-8, 381–382 (1962)

    Article  Google Scholar 

  6. Gong, G., Song, H.-Y.: Two-tuple balance of non-binary sequences with ideal two-level autocorrelation. Discret. Appl. Math. 154(18), 2590–2598 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Helleseth, T., Gong, G.: New nonbinary sequences with ideal two-level autocorrelation. IEEE Trans. Inf. Theory 48(11), 2868–2872 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Helleseth, T., Vijay Kumar, P.: Sequences with low correlation. In: Pless, V., Huffman, W.C. (eds.) Handbook of Coding Theory, vol. II. Elsevier (1998)

  9. Høholdt, T., Justesen, J.: Ternary sequences with perfect periodic autocorrelation (Corresp). IEEE Trans. Inf. Theory 29(4), 597–600 (1983)

    Article  MATH  Google Scholar 

  10. Ipatov, V.P.: Ternary sequences with ideal periodic autocorrelation properties. Radio Eng. Electron. Phys. 24, 75–79 (1979)

    Google Scholar 

  11. Ipatov, V.P.: Multiphase sequences spectrums, Izvestiya VUZ. Radioelektronika (Radioelectronics and Communications Systems) 22(9), 80–82 (1979)

    Google Scholar 

  12. Ipatov, V.P.: Contribution to the theory of sequences with perfect periodic autocorrelation properties. Radio Eng. Electron. Phys. 25, 31–34 (1980)

    MathSciNet  Google Scholar 

  13. Ipatov, V.P.: Spread Spectrum and CDMA: Principles and Applications. Wiley (2005)

  14. Jungnickel, D., Pott, A.: Perfect and Almost Perfect Sequences. Discret. Appl. Math. 95, 331–359 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kumar, T., Helleseth, P., Martinsen, H.: Vijay A new family of ternary sequences with ideal two-level autocorrelation function. Des. Codes Crypt. 23, 157–166 (2001)

    Article  MATH  Google Scholar 

  16. Lee, C.M.: On a new class of 5-ary sequences exhibiting ideal periodic autocorrelation properties with applications to spread spectrum systems. PhD thesis, Department of Electrical Engineering Mississipi State University (1986)

  17. Parraud, P.: On the Non-existence of (Almost-)Perfect Quaternary Sequences. Lect. Notes Comput. Sci 2227, 210–218 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Popovic, B.M.: Generalized chirp-like polyphase sequences with optimum correlation properties. IEEE Trans. Inf. Theory 38(4), 1406–1409 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  19. Simon, M.K., Omura, J.K., Scholtz, R.A., Levitt, B.K.: Spread Spectrum Communications Handbook. McGrawHill (1994)

  20. Udaya, P., Siddiqi, M.U.: Optimal and suboptimal quadriphase sequences derived from maximal length sequences over \(\mathbb {Z}_{4}\). J. Appl. Algebra Eng. Commun. 9, 161–191 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zierler, N.: Linear Recurring Sequences. SIAM J. 7(1), 31–48 (1959)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

Serdar Boztaş was partially supported by an Australian Department of Defence grant. Eda Tekin was supported by a grant from The Scientific and Technological Research Council of Turkey BIDEB 2214-A Program to visit RMIT University where part of this work was performed. We thank the reviewers for their suggestions which greatly improved the presentation of the paper.

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Authors and Affiliations

  1. Discipline of Mathematics, School of Science RMIT University, GPO Box 2476V, Melbourne, VIC, 3001, Australia

    Serdar Boztaş

  2. Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey

    Ferruh Özbudak

  3. Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey

    Ferruh Özbudak & Eda Tekin

  4. Department of Computer Technologies, Karabük University, 78050, Karabük, Turkey

    Eda Tekin

Authors
  1. Serdar Boztaş
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  2. Ferruh Özbudak
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  3. Eda Tekin
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Corresponding author

Correspondence to Serdar Boztaş.

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This article is part of the Topical Collection on Sequences and Their Applications.

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Boztaş, S., Özbudak, F. & Tekin, E. Generalized nonbinary sequences with perfect autocorrelation, flexible alphabets and new periods. Cryptogr. Commun. 10, 509–517 (2018). https://doi.org/10.1007/s12095-017-0237-6

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  • DOI: https://doi.org/10.1007/s12095-017-0237-6

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